Finite Group DatabasebetaFinite Group Databasebeta| order | 336 |
| isoclinism | isoclinic class of ΓL(1,𝔽8)⋊Q8 |
| automorphism | Aut(G) = C24×AGL(1,𝔽7) (order 672) · Inn(G) = C2×AGL(1,𝔽7) · Out(G) = C23 |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | (ΓL(1,𝔽8)⋊C4)⋊C4 |
| unambiguous semidirect product | (Dic7⋊C4)⋊C3 |
| Hall | Dic7⋊C4 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ C4⋊C4 |
| Sylow | C7 ⋊ C3×C4⋊C4 |
52 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 112 |
A minimal faithful permutation representation — 2 generators on 15 points:
(1,2,7)(3,4,6)(8,9,10,11)(13,15);(2,3)(4,6)(5,7)(12,13,14,15)Polycyclic presentation:
2^2,3,2^2,7|a^2=e,b^2=d,b^a=be,f^a=f^6,f^c=f^438 conjugacy classes — view the full character table.
| maximal subgroup of | C4×ΓL(1,𝔽8)⋊Q8, #6433⋊C3, #6435⋊C3, … |
| maximal normal subgroup of | C4×ΓL(1,𝔽8)⋊Q8, #6433⋊C3, #6435⋊C3, … |
| maximal quotient of | #6381⋊C3, #6382⋊C3, (ΓL(1,𝔽8)⋊C4)⋊C8, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(336, 15);